existence of the conditional expectation
Let be a probability space and be a random variable. For any -algebra , we show the existence of the conditional expectation . Although it is possible to do this using the Radon-Nikodym theorem, a different approach is used here which relies on the completeness of the vector space . The defining property of the conditional expectation is
for sets . We shall prove the existence of the conditional expectation for all nonnegative random variables and, more generally, whenever is almost surely finite.
First, the conditional expectation of every square-integrable random variable exists.
Consider the norm on the vector space of real valued random variables satisfying (up to almost everywhere equivalence). This is given by the following inner product
By the existence of orthogonal projections (http://planetmath.org/ProjectionsAndClosedSubspaces) onto closed subspaces of Hilbert spaces, there is an orthogonal projection . In particular, for all and . Setting gives
as required. ∎
We can now prove the existence of conditional expectations of nonnegative random variables. Note that here there are no integrability conditions on .
Let be a nonnegative random variable taking values in . Then, there exists a nonnegative -measurable random variable taking values in and satisfying (1) for all . Furthermore, is uniquely defined -almost everywhere (http://planetmath.org/AlmostSurely).
So is a nonnegative random variable with nonpositive expectation, hence is almost surely equal to zero. Therefore, (almost surely) and, by replacing with the maximum of we may suppose that is an increasing sequence of random variables. Setting , the monotone convergence theorem gives
Finally, we show existence of the conditional expectation of every random variable satisfying almost surely. Note, in particular, that this is satisfied whenever is integrable, as
Let be a random variable such that almost surely. Then, there exists a -measurable random variable such that and (1) is satisfied for every with .
Furthermore, is uniquely defined up to -a.e. equivalence.
The positive and negative parts of satisfy
almost surely. We can therefore set and .
If satisfies then , so and,
Finally, suppose that satisfies the same conditions as . For any set . Then,
So, and are finite, hence (1) gives
So and, letting increase to infinity, almost surely. Similarly, and therefore almost surely. ∎
|Title||existence of the conditional expectation|
|Date of creation||2013-03-22 18:39:28|
|Last modified on||2013-03-22 18:39:28|
|Last modified by||gel (22282)|